Two-impurity scattering in one-dimensional interacting fermion systems

Using eigenfunctional theory, we study two-impurity scattering in one-dimensional interacting fermion systems, and clearly show the evolution of the systems from ultraviolet fixed point to infrared critical fixed point as increasing the impurity scattering potential. For the single-impurity scattering problem, we obtain the same fermion Green's function as that by previous methods. For two-impurity scattering, we demonstrate that the nonlocal interaction of the fermions produced by impurity scattering depends on not only the scattering potential, but also the distance between these two impurities. As two impurities far away from each other, the fermions are completely reflected by the impurities, and in the region between these two impurities there appear stationary waves; while as these two impurities are very close, the fermions with some discrete momentums can easily transmit through them.